Add one tbs of olive oil to a Dutch oven. Brown the turkey sausage in the Dutch oven and break it up with a wooden spoon as it cooks until it is no longer pink, about 5-7 minutes. Remove the sausage with a slotted spoon and set aside.
Add the remaining one tablespoon of olive oil to the Dutch oven and add the onion and carrots. Sauté until the vegetables are fork-tender, about five minutes. If you want to sub minced garlic for the roasted garlic, add it now and saute until fragrant, about 1 minute. If using roasted garlic, squeeze the roasted garlic cloves into the vegetable mixture and stir to combine.
Add the bone broth, potatoes, and thyme to the pot, cover with a lid, and bring to a low simmer. Cook until the potatoes are tender, about 30 minutes.
When the vegetables are cooked through, remove the thyme and use an immersion blender to blend the veggies until the soup is smooth and thick. You can also ladle the soup into a standard blender, mash with a potato masher, or omit this step for a more rustic soup.
Return the sausage to the pot, add half-and-half or coconut milk, and simmer for another 5-10 minutes. Top each bowl of soup with Parmesan cheese.
Enjoy!